Chan, Shu-Park “Section I – Circuits” The Electrical Engineering Handbook Ed. Richard C. Dorf Boca Raton: CRC Press LLC, 2000 The Intel Pentium® processor, introduced at speeds of up to 300 MHz, combines the architectural advances in the Pentium Pro processor with the instruction set extensions of Intel MMX™ media enhancement tech- nology. This combination delivers new levels of performance and the fastest Intel processor to workstations. The Pentium II processor core, with 7.5 million transistors, is based on Intel’s advanced P6 architecture and is manufactured on .35-micron process technology. First implemented in the Pentium Pro processor, the Dual Independent Bus architecture is made up of the L2 cache bus and the processor-to-main-memory system bus. The latter enables simultaneous parallel transactions instead of single, sequential transactions of previous generation processors. The types of applications that will benefit from the speed of the Pentium II processor and the media enhancement of MMX technology include scanning, image manipulation, video conferencing, Internet browsers and plug-ins, video editing and playback, printing, faxing, compression, and encryption. The Pentium II processor is the newest member of the P6 processor family, but certainly not the last in the line of high performance processors. (Courtesy of Intel Corporation.) © 2000 by CRC Press LLC © 2000 by CRC Press LLC I Circuits 1 Passive Components M. Pecht, P. Lall, G. Ballou, C. Sankaran, N. Angelopoulos Resistors • Capacitors and Inductors • Transformers • Electrical Fuses 2 Voltage and Current Sources R.C. Dorf, Z. Wan, C.R. Paul, J.R. Cogdell Step, Impulse, Ramp, Sinusoidal, Exponential, and DC Signals • Ideal and Practical Sources • Controlled Sources 3 Linear Circuit Analysis M.D. Ciletti, J.D. Irwin, A.D. Kraus, N. Balabanian, T.A. Bickart, S.P. Chan, N.S. Nise Voltage and Current Laws • Node and Mesh Analysis • Network Theorems • Power and Energy • Three-Phase Circuits • Graph Theory • Two Port Parameters and Transformations 4 Passive Signal Processing W.J. Kerwin Low-Pass Filter Functions • Low-Pass Filters • Filter Design 5 Nonlinear Circuits J.L. Hudgins, T.F. Bogart, Jr., K. Mayaram, M.P. Kennedy, G. Kolumbán Diodes and Rectifiers • Limiters • Distortion • Communicating with Chaos 6 Laplace Transform R.C. Dorf, Z. Wan, D.E. Johnson Definitions and Properties • Applications 7 State Variables: Concept and Formulation W.K. Chen State Equations in Normal Form • The Concept of State and State Variables and Normal Tree • Systematic Procedure in Writing State Equations • State Equations for Networks Described by Scalar Differential Equations • Extension to Time-Varying and Nonlinear Networks 8 The z-Transform R.C. Dorf, Z. Wan Properties of the z -Transform • Unilateral z -Transform • z -Transform Inversion • Sampled Data 9T- P Equivalent Networks Z. Wan, R.C. Dorf Three-Phase Connections • Wye ⇔ Delta Transformations 10 Transfer Functions of Filters R.C. Dorf, Z. Wan Ideal Filters • The Ideal Linear-Phase Low-Pass Filter • Ideal Linear-Phase Bandpass Filters • Causal Filters • Butterworth Filters • Chebyshev Filters 11 Frequency Response P. Neudorfer Linear Frequency Response Plotting • Bode Diagrams • A Comparison of Methods 12 Stability Analysis F. Szidarovszky, A.T. Bahill Using the State of the System to Determine Stability • Lyapunov Stability Theory • Stability of Time-Invariant Linear Systems • BIBO Stability • Physical Examples 13 Computer Software for Circuit Analysis and Design J.G. Rollins, P. Bendix Analog Circuit Simulation • Parameter Extraction for Analog Circuit Simulation © 2000 by CRC Press LLC Shu-Park Chan International Technological University HIS SECTION PROVIDES A BRIEF REVIEW of the definitions and fundamental concepts used in the study of linear circuits and systems. We can describe a circuit or system , in a broad sense, as a collection of objects called elements ( components, parts, or subsystems ) which form an entity governed by certain laws or constraints. Thus, a physical system is an entity made up of physical objects as its elements or components. A subsystem of a given system can also be considered as a system itself. A mathematical model describes the behavior of a physical system or device in terms of a set of equations, together with a schematic diagram of the device containing the symbols of its elements, their connections, and numerical values. As an example, a physical electrical system can be represented graphically by a network which includes resistors, inductors, and capacitors, etc. as its components. Such an illustration, together with a set of linear differential equations, is referred to as a model system. Electrical circuits may be classified into various categories. Four of the more familiar classifications are (a) linear and nonlinear circuits, (b) time-invariant and time-varying circuits, (c) passive and active circuits, and (d) lumped and distributed circuits. A linear circuit can be described by a set of linear (differential) equations; otherwise it is a nonlinear circuit. A time-invariant circuit or system implies that none of the components of the circuit have parameters that vary with time; otherwise it is a time-variant system. If the total energy delivered to a given circuit is nonnegative at any instant of time, the circuit is said to be passive ; otherwise it is active . Finally, if the dimensions of the components of the circuit are small compared to the wavelength of the highest of the signal frequencies applied to the circuit, it is called a lumped circuit; otherwise it is referred to as a distributed circuit. There are, of course, other ways of classifying circuits. For example, one might wish to classify circuits according to the number of accessible terminals or terminal pairs (ports). Thus, terms such as n-terminal circuit and n-port are commonly used in circuit theory. Another method of classification is based on circuit configu- rations (topology), 1 which gives rise to such terms as ladders, lattices, bridged-T circuits, etc. As indicated earlier, although the words circuit and system are synonymous and will be used interchangeably throughout the text, the terms circuit theory and system theory sometimes denote different points of view in the study of circuits or systems. Roughly speaking, circuit theory is mainly concerned with interconnections of components (circuit topology) within a given system, whereas system theory attempts to attain generality by means of abstraction through a generalized (input-output state) model. One of the goals of this section is to present a unified treatment on the study of linear circuits and systems. That is, while the study of linear circuits with regard to their topological properties is treated as an important phase of the entire development of the theory, a generality can be attained from such a study. The subject of circuit theory can be divided into two main parts, namely, analysis and synthesis. In a broad sense, analysis may be defined as “the separating of any material or abstract entity [system] into its constituent elements;” on the other hand, synthesis is “the combining of the constituent elements of separate materials or abstract entities into a single or unified entity [system].” 2 It is worth noting that in an analysis problem, the solution is always unique no matter how difficult it may be, whereas in a synthesis problem there might exist an infinite number of solutions or, sometimes, none at all! It should also be noted that in some network theory texts the words synthesis and design might be used interchangeably throughout the entire discussion of the subject. However, the term synthesis is generally used to describe analytical procedures that can usually be carried out step by step, whereas the term design includes practical (design) procedures (such as trial-and-error techniques which are based, to a great extent, on the experience of the designer) as well as analytical methods. In analyzing the behavior of a given physical system, the first step is to establish a mathematical model. This model is usually in the form of a set of either differential or difference equations (or a combination of them), 1 Circuit topology or graph theory deals with the way in which the circuit elements are interconnected. A detailed discussion on elementary applied graph theory is given in Chapter 3.6. 2 The definitions of analysis and synthesis are quoted directly from The Random House Dictionary of the English Language, 2nd ed., Unabridged, New York: Random House, 1987. T © 2000 by CRC Press LLC the solution of which accurately describes the motion of the physical systems. There is, of course, no exception to this in the field of electrical engineering. A physical electrical system such as an amplifier circuit, for example, is first represented by a circuit drawn on paper. The circuit is composed of resistors, capacitors, inductors, and voltage and/or current sources, 1 and each of these circuit elements is given a symbol together with a mathe- matical expression (i.e., the voltage-current or simply v-i relation) relating its terminal voltage and current at every instant of time. Once the network and the v-i relation for each element is specified, Kirchhoff’s voltage and current laws can be applied, possibly together with the physical principles to be introduced in Chapter 3.1, to establish the mathematical model in the form of differential equations. In Section I, focus is on analysis only (leaving coverage of synthesis and design to Section III, “Electronics”). Specifically, the passive circuit elements—resistors, capacitors, inductors, transformers, and fuses—as well as voltage and current sources (active elements) are discussed. This is followed by a brief discussion on the elements of linear circuit analysis. Next, some popularly used passive filters and nonlinear circuits are introduced. Then, Laplace transform, state variables, z -transform, and T and p configurations are covered. Finally, transfer functions, frequency response, and stability analysis are discussed. Nomenclature 1 Here, of course, active elements such as transistors are represented by their equivalent circuits as combinations of resistors and dependent sources. Symbol Quantity Unit A area m 2 B magnetic flux density Tesla C capacitance F e induced voltage V e dielectric constant F/m e ripple factor f frequency Hz F force Newton f magnetic flux weber I current A J Jacobian k Boltzmann constant 1.38 ´ 10 –23 J/K k dielectric coefficient K coupling coefficient L inductance H l eigenvalue M mutual inductance H n turns ratio n filter order Symbol Quantity Unit w angular frequency rad/s P power W PF power factor q charge C Q selectivity R resistance W R ( T ) temperature coefficient W / ° C of resistance r resistivity W m s Laplace operator t damping factor q phase angle degree v velocity m/s V voltage V W energy J X reactance W Y admittance S Z impedance W Pecht, M., Lall, P., Ballou, G., Sankaran, C., Angelopoulos, N. “Passive Components” The Electrical Engineering Handbook Ed. Richard C. Dorf Boca Raton: CRC Press LLC, 2000 © 2000 by CRC Press LLC 1 Passive Components 1.1 Resistors Resistor Characteristics•Resistor Types 1.2 Capacitors and Inductors Capacitors•Types of Capacitors•Inductors 1.3 Transformers Types of Transformers•Principle of Transformation•Electromagnetic Equation•Transformer Core•Transformer Losses•Transformer Connections•Transformer Impedance 1.4 Electrical Fuses Ratings•Fuse Performance•Selective Coordination•Standards•Products•Standard—Class H• HRC•Trends 1.1 Resistors Michael Pecht and Pradeep Lall The resistor is an electrical device whose primary function is to introduce resistance to the flow of electric current. The magnitude of opposition to the flow of current is called the resistance of the resistor. A larger resistance value indicates a greater opposition to current flow. The resistance is measured in ohms. An ohm is the resistance that arises when a current of one ampere is passed through a resistor subjected to one volt across its terminals. The various uses of resistors include setting biases, controlling gain, fixing time constants, matching and loading circuits, voltage division, and heat generation. The following sections discuss resistor characteristics and various resistor types. Resistor Characteristics Voltage and Current Characteristics of Resistors The resistance of a resistor is directly proportional to the resistivity of the material and the length of the resistor and inversely proportional to the cross-sectional area perpendicular to the direction of current flow. The resistance R of a resistor is given by (1.1) where r is the resistivity of the resistor material ( W · cm), l is the length of the resistor along direction of current flow (cm), and A is the cross-sectional area perpendicular to current flow (cm 2 ) (Fig. 1.1). Resistivity is an inherent property of materials. Good resistor materials typically have resistivities between 2 ´ 10 –6 and 200 ´ 10 –6 W · cm. R l A = r Michael Pecht University of Maryland Pradeep Lall Motorola Glen Ballou Ballou Associates C. Sankaran Electro-Test Nick Angelopoulos Gould Shawmut Company © 2000 by CRC Press LLC The resistance can also be defined in terms of sheet resistivity. If the sheet resistivity is used, a standard sheet thickness is assumed and factored into resistivity. Typically, resistors are rectangular in shape; therefore the length l divided by the width w gives the number of squares within the resistor (Fig. 1.2). The number of squares multiplied by the resistivity is the resistance. (1.2) where r sheet is the sheet resistivity ( W /square), l is the length of resistor (cm), w is the width of the resistor (cm), and R sheet is the sheet resistance ( W ). The resistance of a resistor can be defined in terms of the voltage drop across the resistor and current through the resistor related by Ohm’s law, (1.3) where R is the resistance ( W ), V is the voltage across the resistor (V), and I is the current through the resistor (A). Whenever a current is passed through a resistor, a voltage is dropped across the ends of the resistor. Figure 1.3 depicts the symbol of the resistor with the Ohm’s law relation. All resistors dissipate power when a voltage is applied. The power dissipated by the resistor is represented by (1.4) where P is the power dissipated (W), V is the voltage across the resistor (V), and R is the resistance ( W ). An ideal resistor dissipates electric energy without storing electric or magnetic energy. Resistor Networks Resistors may be joined to form networks. If resistors are joined in series, the effective resistance ( R T ) is the sum of the individual resistances (Fig. 1.4). (1.5) FIGURE 1.1Resistance of a rectangular cross-section resistor with cross-sectional area A and length L. R l w sheet sheet =r R V I = P V R = 2 RR Ti i n = = å 1 FIGURE 1.2Number of squares in a rectangular resistor. FIGURE 1.3A resistor with resistance R having a current I flowing through it will have a voltage drop of IR across it. © 2000 by CRC Press LLC If resistors are joined in parallel, the effective resistance ( R T ) is the reciprocal of the sum of the reciprocals of individual resistances (Fig. 1.5). (1.6) Temperature Coefficient of Electrical Resistance The resistance for most resistors changes with temperature. The tem- perature coefficient of electrical resistance is the change in electrical resistance of a resistor per unit change in temperature. The tempera- ture coefficient of resistance is measured in W / ° C. The temperature coefficient of resistors may be either positive or negative. A positive temperature coefficient denotes a rise in resistance with a rise in tem- perature; a negative temperature coefficient of resistance denotes a decrease in resistance with a rise in temperature. Pure metals typically have a positive temperature coefficient of resistance, while some metal alloys such as constantin and manganin have a zero temperature coef- ficient of resistance. Carbon and graphite mixed with binders usually exhibit negative temperature coefficients, although certain choices of binders and process variations may yield positive temperature coeffi- cients. The temperature coefficient of resistance is given by R ( T 2 ) = R ( T 1 )[1 + a T 1 ( T 2 – T 1 )] (1.7) where a T 1 is the temperature coefficient of electrical resistance at reference temperature T 1 , R ( T 2 ) is the resistance at temperature T 2 ( W ), and R ( T 1 ) is the resistance at temperature T 1 ( W ). The reference temperature is usually taken to be 20 ° C. Because the variation in resistance between any two temperatures is usually not linear as predicted by Eq. (1.7), common practice is to apply the equation between temperature increments and then to plot the resistance change versus temperature for a number of incremental temperatures. High-Frequency Effects Resistors show a change in their resistance value when subjected to ac voltages. The change in resistance with voltage frequency is known as the Boella effect. The effect occurs because all resistors have some inductance and capacitance along with the resistive component and thus can be approximated by an equivalent circuit shown in Fig. 1.6. Even though the definition of useful frequency range is application dependent, typically, the useful range of the resistor is the highest frequency at which the impedance differs from the resistance by more than the tolerance of the resistor. The frequency effect on resistance varies with the resistor construction. Wire-wound resistors typically exhibit an increase in their impedance with frequency. In composition resistors the capacitances are formed by the many conducting particles which are held in contact by a dielectric binder. The ac impedance for film resistors remains constant until 100 MHz (1 MHz = 10 6 Hz) and then decreases at higher frequencies (Fig. 1.7). For film resistors, the decrease in dc resistance at higher frequencies decreases with increase in resistance. Film resistors have the most stable high-frequency performance. FIGURE 1.4 Resistors connected in series. 11 1 RR Ti i n = = å FIGURE 1.5Resistors connected in parallel. FIGURE 1.6Equivalent circuit for a resistor. © 2000 by CRC Press LLC The smaller the diameter of the resistor the better is its frequency response. Most high-frequency resistors have a length to diameter ratio between 4:1 to 10:1. Dielectric losses are kept to a minimum by proper choice of base material. Voltage Coefficient of Resistance Resistance is not always independent of the applied voltage. The voltage coefficient of resistance is the change in resistance per unit change in voltage, expressed as a percentage of the resistance at 10% of rated voltage. The voltage coefficient is given by the relationship (1.8) where R 1 is the resistance at the rated voltage V 1 and R 2 is the resistance at 10% of rated voltage V 2 . Noise Resistors exhibit electrical noise in the form of small ac voltage fluctuations when dc voltage is applied. Noise in a resistor is a function of the applied voltage, physical dimensions, and materials. The total noise is a sum of Johnson noise, current flow noise, noise due to cracked bodies, and loose end caps and leads. For variable resistors the noise can also be caused by the jumping of a moving contact over turns and by an imperfect electrical path between the contact and resistance element. The Johnson noise is temperature-dependent thermal noise (Fig. 1.8). Thermal noise is also called “white noise” because the noise level is the same at all frequencies. The magnitude of thermal noise, E RMS (V), is dependent on the resistance value and the temperature of the resistance due to thermal agitation. (1.9) where E RMS is the root-mean-square value of the noise voltage (V), R is the resistance ( W ), K is the Boltzmann constant (1.38 ´ 10 –23 J/K), T is the temperature (K), and Df is the bandwidth (Hz) over which the noise energy is measured. Figure 1.8 shows the variation in current noise versus voltage frequency. Current noise varies inversely with frequency and is a function of the current flowing through the resistor and the value of the resistor. The magnitude of current noise is directly proportional to the square root of current. The current noise magnitude is usually expressed by a noise index given as the ratio of the root-mean-square current noise voltage (E RMS ) FIGURE 1.7Typical graph of impedance as a percentage of dc resistance versus frequency for film resistors. Voltage coefficient= 100( ( 1 21 RR RVV –) –) 2 2 E kRTf RMS = 4 D [...]... ranging from 1/8 to 2 W Composition resistors have end-to-end shunted capacitance that may be noticed at frequencies in the neighborhood of 100 kHz, especially for resistance values above 0.3 MW Metal-Film Resistors Metal-lm resistors are commonly made of nichrome, tin-oxide, or tantalum nitride, either hermetically sealed or using molded-phenolic cases Metal-lm resistors are not as stable as the TABLE 1.2... for current-limiting or voltage-dropping functions in circuit applications Semiprecision resistors have long-term temperature stability General-purpose resistors are used in circuits that do not require tight resistance tolerances or long-term stability For general-purpose resistors, initial resistance variation may be in the neighborhood of 5% and the variation in resistance under full-rated power... resistors Thin-lm Tantalum SnO2 Ni-Cr Cermet (Cr-SiO) Thick-lm Ruthenium-silver Palladium-silver Sheet Resistivity (per square) Temperature Coefcient (ppm/C) 0.8 to 260 W 0.003 to 10 kW 0.001 to 10 kW 0.5 to 20 kW 1100 to 2000 2900 to 5000 3000 to 6000 100 to 1300 0.01 to 1 kW 0.08 to 4 kW 40 to 450 W 0.03 to 2.5 kW 10 W to 10 MW 0.01 to 100 kW m100 1500 to 0 m100 m150 m200 500 to 150 Ion-implanted resistors... of adding dopants is called doping Dopants may belong to group IIIA (e.g., boron) or group VA (e.g., phosphorus) in the periodic table If the elements belong to the group IIIA, the resulting semiconductor is called a p-type semiconductor On the other hand, if the elements belong to the group VA, the resulting semiconductor is called an n-type semiconductor Epitaxial layer: Epitaxy refers to processes... core due to the excitation current i: FIGURE 1.24 Electromagnetic relation â 2000 by CRC Press LLC ổ pử f = F sin ỗ wt - ữ = - F cos(wt ) 2ứ ố Induced voltage in the winding e = -N df d [F cos(wt )] = N = -N w F sin(wt ) dt dt Maximum value of the induced voltage E = NwF The root-mean-square value E rms = E 2 = 2 pf N F = 4.44 f NBA 2 where ux F (webers) is replaced by the product of the ux density B... classied into thick-lm and thin-lm resistors Semiconductor resistors can be divided into four types: diffused, bulk, pinched, and ion-implanted Table 1.3 shows some typical resistor properties for semiconductor resistors Diffused semiconductor resistors use resistivity of the diffused region in the semiconductor substrate to introduce a resistance in the circuit Both n-type and p-type diffusions are... Mica capacitors have small capacitance values and are usually used in high-frequency circuits They are constructed as alternate layers of metal foil and mica insulation, which are stacked and encapsulated, or are silvered mica, where a silver electrode is screened on the mica insulators Paper-Foil-Filled Capacitors Paper-foil-lled capacitors are often used as motor capacitors and are rated at 60 Hz... encasements Wet-electrolyte sintered-anode tantalum capacitors, often called wet-slug tantalum capacitors, use a pellet of sintered tantalum powder to which a lead has been attached, as shown in Fig 1.19 This anode has an enormous surface area for its size Wet-slug tantalum capacitors are manufactured in a voltage range to 125 V dc Use Considerations Foil tantalum capacitors are used only where high-voltage... special-purpose Each of these resistor types is discussed in detail with typical ranges of their characteristics Fixed Resistors The xed resistors are those whose value cannot be varied after manufacture Fixed resistors are classied into composition resistors, wire-wound resistors, and metal-lm resistors Table 1.2 outlines the characteristics of some typical xed resistors Wire-Wound Resistors Wire-wound... high-energy ions The implanted ions lie in a very shallow layer along the surface (0.1 to 0.8 mm) For similar thicknesses ion-implanted resistors yield sheet resistivities 20 times greater than diffused resistors Table 1.3 shows typical properties of diffused, bulk, pinched, and ion-implanted resistors Typical sheet resistance values range from 80 to 250 W/square Varistors Varistors are voltage-dependent . Extension to Time-Varying and Nonlinear Networks 8 The z-Transform R.C. Dorf, Z. Wan Properties of the z -Transform • Unilateral z -Transform • z -Transform. performance and the fastest Intel processor to workstations. The Pentium II processor core, with 7.5 million transistors, is based on Intel s advanced P6 architecture